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The link to the article is (https://arxiv.org/abs/1103.5728)[https://arxiv.org/abs/1103.5728] .

    I think, the following code is relevant - for the comment, not for the posted question:

p0 = 5
n  = 7

S.<x> = ZZ[]

R  =   x^n     - p0^(n-1)*x + p0
Rx = n*x^(n-1) - p0^(n-1)

F.<u> = CyclotomicField( n-1 )
L.<a> = F.extension( x^(n-1) - n )

LPrimes = L.primes_of_degree_one_list( 100 )

for LP1 in LPrimes:
    p1  = LP1.relative_norm().norm()
    Fp1 = GF(p1)
    Rp1 = PolynomialRing( Fp1, names='x' )
    print "p1 = %s" % p1
    print "R  mod p1 is %s :: %s" % ( Rp1(R ), 'IRRED' if Rp1(R).is_irreducible() else 'RED' )
    print "Rx mod p1 is %s == %s" % ( Rp1(Rx), Rp1(Rx).factor() )

(Have to send, and catch that train, possibly be back again.)